BMS charges in polyhomogeneous spacetimes

We classify the asymptotic charges of a class of polyhomogeneous asymptotically-flat spacetimes with finite shear, generalising recent results on smooth asymptotically-flat spacetimes. Polyhomogenous spacetimes are a formally consistent class of spacetimes that do not satisfy the well-known peeling property. As such, they constitute a more physical class of asymptotically-flat spacetimes compared to the smooth class. In particular, we establish that the generalised conserved non-linear Newman-Penrose charges that are known to exist for such spacetimes are a subset of asymptotic BMS charges.

One of the most striking results in the mathematical study of gravitational waves in general relativity is the so-called peeling property [1][2][3] (see, e.g., also Ref. [4]). The peeling property is a statement regarding the asymptotic behaviour of the Weyl tensor components as one approaches null infinity. For a smooth asymptotically-flat spacetime, the result follows from the assumed smoothness of the unphysical spacetime upon conformal compactification [5]. In Bondi coordinates [1,2], it is where r is an affine parameter along an outgoing null geodesic. The superscripts on the Weyl tensors on the RHS denote the components of the Weyl tensor in a null basis that is used to define the Petrov type of the spacetime. Thus, the leading order term corresponds to the Weyl tensor components of Petrov type N. Given that the Weyl tensor encompasses the remaining degrees of freedom in the curvature, the peeling property can be viewed as a statement regarding the fall-off behaviour of isolated gravitating bodies and the radiation they emit. However, already in Ref. [3] (see footnote 27), doubts were expressed regarding the validity of the assumptions that lead to this result. Since then, the question of the validity of the peeling property has attracted much research; see e.g. Refs. [6][7][8][9][10][11][12][13][14][15][16]. What is clear by now is that the smoothness assumption precludes many interesting physically relevant cases. One class of spacetimes that move away from the smoothness assumption are those that admit a polyhomogeneous expansion [12]. These spacetimes are formally consistent with the Einstein equations and admit the BMS group as an asymptotic symmetry group [12], have a welldefined Trautman-Bondi mass parameter [17] and admit Newman-Penrose charges [18,19].
Importantly, they provide an example of a more realistic class of asymptotically-flat spacetimes than the smooth case.
In this paper, we shall study the asymptotic BMS charges admitted by polyhomogeneous spacetimes that have a finite shear [18]. This subset of polyhomogeneous spacetimes have a slightly better fall-off property at leading order compared with the most general spacetimes.
We will concentrate on this large subset of polyhomogeneous spacetimes in order to make the rather involved calculations tractable. However, we are confident that the results obtained in this paper may be generalised to the full class.
In recent work, a relation has been established [20,21], in the smooth case, between Newman-Penrose charges [22], which are a set of ten conserved non-linear charges at null infinity, and asymptotic BMS charges [23][24][25], which are the charges associated with the gen-erators of the BMS algebra via the Noether theorem. Although, such a relation ought to be natural, remarkably, such a relation had not been previously found. Indeed, in order to make progress, it has been required to extend the notion of asymptotic BMS charges to include subleading charges [20] and new dual charges [21,26], which have recently been derived from first principles [27,28].
Our aim in this work is to extend the formalism developed in Refs. [20,21] to classify the asymptotic BMS charges within the class of polyhomogeneous spacetimes with finite shear.
This generalisation is non-trivial for two reasons: the calculational complexity increases substantially when considering polyhomogenous spacetimes and particular features of the polyhomogeneous expansion raises interesting new questions about the nature of charges, as will become apparent below. In particular, the nature of the characteristic value problem applied to polyhomogeneous spacetimes means that non-trivial conserved BMS charges can be defined in terms of initial data that do not evolve. This is a new feature that is specific to polyhomogeneous spacetimes and compels us to reappraise what we mean by conserved BMS charges.
Our main result is to establish a relation between asymptotic BMS charges and the generalised Newman-Penrose charges discovered in Ref. [18] for polyhomogenous spacetimes with finite shear. Generalised Newman-Penrose charges exist for the full class of polyhomogeneous spacetimes [19] and we expect this relation to also hold in the full class.
The insights gained from this study have led to a better understanding of how Newman-Penrose charges come about and the possibility of identifying conserved charges at lower orders.
An as of yet open question is whether conserved charges could exist at lower orders in the 1/rexpansion. We hope to tackle this interesting problem in a future work.
In Section 2, we give some prerequisite information regarding polyhomogeneous spacetimes and the fall-off behaviour of their Weyl tensor components, the Einstein equations and the action of the BMS group on the metric components. Also, we define the subclass of polyhomogenous spacetimes with finite shear. In Section 3, we classify the standard BMS charges up to order 1/r 3 and identify a subset of five conserved non-linear charges. Similarly, in Section 4, we classify the dual charges defined in Ref. [21] up to order 1/r 3 and, again, discover a subset of five conserved non-linear charges. In Section 5, we show, via a translation to the Newman-Penrose formalism, that the set of ten conserved charges found in Sections 3 and 4 are equivalent to the generalised Newman-Penrose charges of Ref. [18].
A polyhomogeneous spacetime is one for which the metric components can be expanded asymptotically as a combination of powers of r −1 and positive powers of z ≡ log r as r → ∞. For example, a function f admits a polyhomogeneous expansion if where each f i is itself a series expansion in positive powers of z. As in Ref. [12,19], we restrict our attention to spacetimes where only finite powers of z appear in the series, so that f i are polynomials in z. 1 Following Ref. [19], we denote the degree of a polynomial f as #f .
Working with the Bondi definition of asymptotic flatness [1,2], we introduce Bondi coordinates (u, r, x I ) with x I = {θ, φ}, such that the metric takes the form where a residual gauge freedom in defining r is fixed by imposing with ω IJ the standard metric on the round 2-sphere. This condition implies that h IJ has two degrees of freedom.
The Bondi definition of asymptotic flatness and the condition of polyhomogeneity requires that the metric parameters have the following large r asymptotic form 2 β(u, r, x I ) = β 0 (z, u, x I ) 1 Relaxing this condition would mean that the infinite series in z that would appear in these calculations would in fact reduce to integer powers of r. Therefore, our analysis, which treats the expansions in 1/r and z independently, would no longer be valid.

Asymptotic behaviour of Weyl scalar Ψ 0
For spacetimes that are analytic in 1/r, i.e. there exist no log terms, the Weyl tensor satisfies the so-called peeling property [3], which can be simply stated as the fact that the Weyl tensor in the unphysical spacetime vanishes at null infinity. In Newman-Penrose language [3], this statement is equivalent to the fact that where the Ψ i are the Newman-Penrose Weyl scalars defined with respect to a complex null frame (ℓ a , n a , m a ,m a ), As we shall explain below, the peeling property no longer holds in polyhomogeneous spacetimes [12,29]. Moreover, we shall find that the Weyl scalar Ψ 0 falls off too slowly. This will lead us to make some further assumptions on the metric expansion (2.4).
We begin by choosing a complex null frame e µ a = (ℓ a , n a , m a ,m a ) with inverse E µ a , with h IJ the matrix inverse of h IJ . The polyhomogeneous expansion (2.4) implies that Compare this with the fall-off of Ψ 0 in Refs. [20][21][22] In this paper, in order to make progress, we will assume that Ψ 0 behaves asymptotically as O(r −4 log N 4 r). While, it is true that Newman-Penrose charges exist more generally for any polyhomogeneous spacetime defined by the fall-offs (2.4) [19], the analysis is much simpler if we assume that the leading order term in the shear of the null congruence defined by ℓ has no log terms [18]. This is equivalent to the requirement that Ψ 0 ∼ O(r −4 log N 4 r), or that which is equivalent to the condition that B θθ = B θφ = 0, given that B IJ is real. From the fact that B IJ is traceless and symmetric, we deduce that B IJ is traceless and symmetric and hence the above condition is equivalent to The fact that B IJ is a polynomial in z of finite order, implies that i.e. that B IJ is independent of z and contains no log terms. Henceforth, we shall assume that this condition always holds. We shall find below that, together with this condition, the Einstein equations imply that all leading order terms in (2.4) are independent of z. 3 3 At the next order, Ψ0 ∼ 1

Notation
For brevity, it will prove useful to use the following notation λ X(z) ≡ e λz dz e −λz X(z), (2.17) for λ an integer, in order to reduce the size of some of the equations. Furthermore, ∫ λ will be treated as an operator acting on the right, so we have for example For λ = 0, λ does not change the order of the polynomial in z; see Appendix A. However, ∂ z decreases the order by one and 0 increases it by one.
Moreover, angled brackets on pairs of indices will be used to denote the symmetric trace-free part; thus, for an arbitrary tensor X IJ For example, (2.20)

Einstein Equations
We will assume that the energy-momentum tensor satisfies the fall-off conditions 4 The Einstein equation then yields

24)
that Ψ0 = o(r −5 ) would imply that CIJ = 0, which recovers the fall-off conditions (2.2) in Ref. [20]. 4 Given some arbitrary vector Va, we denote the components in the null basis as follows with the obvious generalisation to tensors.
where D I is the standard covariant derivative associated with the round-sphere metric ω IJ .
Since B IJ is independent of z, we conclude the leading order terms in (2.4) are all independent of z.
Assuming stronger fall-off conditions for the energy momentum tensor, the Einstein equations imply the following of Ref. [20]. Setting C IJ = 0 and assuming that all tensors are z-independent, so that equation (A.4) can be used, the above equations reduce to the respective equations in Ref. [20] by taking Assuming the vacuum Einstein equations to the appropriate order, it is possible to deduce the z order of each metric parameter in (2.4). In general, An important assumption that we shall rely upon in what follows is This is the case for generic initial data [19]. It is possible that in special cases, for example if D I C IJ = 0, the above assumption does not hold. Nevertheless, all the charges obtained in this paper are still conserved in such cases.

BMS group
The asymptotic symmetry group of polyhomogeneous spacetimes is given by the BMS group, as with the smooth case [12]. This group is obtained by imposing that the variation of the metric under the generators of the asymptotic symmetry group respects the form of the metric and the gauge choices. These conditions imply a group of the form where ST represents the infinite affine group of supertranslations parameterised by a u and r-independent function s(x I ) and generated by diffeomorphisms of the form As in Ref. [20], we shall concentrate on the supertranslation part of the BMS algebra.
We list below the variation of some of the metric components under supertranslations that will be useful later.
These variations are guaranteed to preserve the form of the metric. However, we will impose further constraints on the metric via the Einstein equations by assuming particular fall-offs of the components of the energy-momentum tensor. If we impose a particular fall-off on one component, we may need to impose further conditions on other components so that the desired fall-off condition is preserved under the BMS action. The variation of a particular component (for fixed α, β ∈ {0, 1, m,m}) is given by If we insist that T αβ = o(r −n ), certain fall-off conditions must be obeyed by T cα and T cβ .
When assuming a particular fall-off condition, we will also assume that the relevant conditions are satisfied for the other components. This can always be done and presents no issues in our calculations.

Standard BMS charges
The asymptotic charges associated with the asymptotic BMS symmetry group is given by the following expression [24] (see also Refs. [23,30]) where we have used the form of the background metric of interest (2.2) in the second equality.

The 2-form H is given by
The slash on the variational symbol δ in (3.1) signifies the fact that the variation is not, in general, integrable.
We have all the ingredients to compute charges, namely the background metric g ab given by (2.2) and the symmetry generators given by (2.40). Plugging the above into equation (3.1) leads to an expansion of the form [20] where each δ /I i (z) is a polynomial of finite order in z = log r. The first term δ /I 0 in the expansion above has been derived previously for smooth asymptotically-flat spacetimes [25]. Below, we find that this result extends to polyhomogeneous spacetimes [17]. Following Ref. [20], we extend the definition of BMS charges to subleading orders in a 1/r-expansion. Investigating these subleading BMS charges in the context of polyhomogeneous spacetimes is indeed the main aim of this paper. We will find that the results in the polyhomogeneous case are analogous to those for smooth spacetimes, albeit, the expressions are rather more complicated.

BMS charge at O(r 0 )
At leading order, we find Observe that at this leading order in the variation of the BMS charges (3.3), we do not encounter log r terms. This is a direct consequence of the finite shear condition (2.16), which implies that all leading order terms in the expansion (2.4) are independent of z.
As in the smooth case [25], the non-integrability above is related to the existence of flux at infinity. In particular, the charge is integrable if and only if ∂ u B IJ = 0, i.e. in the absence of Bondi news at null infinity [31]. The integrable part when integrated over the 2-sphere corresponds to leading-order BMS charges, which generalise the Bondi-Sachs 4-momentum corresponding to s an ℓ = 0 or 1 spherical harmonic.
At the next order, we obtain Therefore, in this case I 1 = 0 and there is no non-trivial charge. If, however, the fall-off of T 01 is weaker, we have non-vanishing charges given by the coefficients of the polynomial in z . It can be shown by considering (2.47) that it is possible to have T mm = o(r −3 ) and T 0m = o(r −4 ) with T 01 non-vanishing at this order. The higher order charges depend only on C IJ . Since we have assumed ∂ u C IJ = 0, such terms are trivially conserved. Therefore, the only interesting charge will be the one corresponding to the z 0 coefficient.
imply equations (2.25), (2.27) and (2.33), the variation of the BMS charge at the next order is As ever, the above separation into the integrable and non-integrable parts is not unique.
The choice above has been made in order to obtain the simplest expressions possible. This will become most clear upon using further Einstein equations. If we further assume that , which imply equations (2.28), (2.31) and (2.34), the above expression reduces to where for brevity, we have not directly substituted the expression for ∂ u D IJ . The integrable piece has z degree N D > N C ≥ 0. A non-trivial charge could appear as a coefficient of each z power in the integrable piece. We first consider the highest order-the coefficient of z N D . The non-integrable piece has maximum z degree N C + 1 as can be seen from (2.38). If N D > N C + 1 then each coefficient of z n for n > N C + 1 in the integrable piece gives a non-trivial charge.
These are where Y ℓm are spherical harmonics. However, inspecting equation (2.34), the Einstein equation for ∂ u D IJ , we notice that the right hand side has z degree N C + 1; hence the higher order terms in D IJ do not evolve, i.e. they are constant in u. Therefore, the fact that the charges defined above are conserved is unsurprising.
The highest non-trivial order to consider is O(z N C +1 ). We must calculate the coefficient Using the results of Appendix A, we find that the terms of z degree N C + 1 in the expression above are of the form / δI For any given s(x), we can choose a C IJ (x) to make the expression in brackets an arbitrary symmetric traceless tensor. That is, for any traceless symmetric X IJ (x) and s(x I ), we can find a traceless symmetric solution C IJ (x) to the second order PDE thus s corresponds to an ℓ = 0 or 1 spherical harmonic (see Appendix C).
From (3.11), we observe that / δI (non−int) 2 C IJ terms vanishes at all orders when s obeys equation (3.14). Moreover, from equation (A.13) in Appendix A, we have that for (3.11) to vanish at a given order, it must vanish at all higher orders, in particular the highest order.
This means that s must obey (3.14) for the coefficients of lower z orders to be integrable.
In conclusion, we deduce that at any order, (3.11) vanishes if and only if s is an ℓ = 0 or 1 spherical harmonic.
Assuming equation (3.14), the non-integrable part of equation (3.9) reduces to / δI which is a total derivative and can, therefore, be ignored. Hence, at all orders in z we obtain the (unintegrated) charges However, up to total derivatives, this is equal to As before, it should be emphasised that the separation into integrable and non-integrable pieces is not unique and the form above has been chosen to make the following expressions simpler.
for n > N D . that can contribute to this order arise from C IJ and D IJ terms. We have where / δI There is no Einstein equation relating D IJ and C IJ , so the contributions from the two terms above, namely / δI (non−int) 3 D terms and / δI (non−int) 3 BC terms need to vanish independently in (3.20) in order for the charge to be integrable in general. We focus on / δI (3.29) The factor sD K D IJ + 5D IJ D K s is an arbitrary tensor that is symmetric and trace-free on Next, we consider the terms in / δI (non−int) 3 BC terms . Reorganising these terms, equation We shall show below in Section 5 that these charges correspond to half of the set of Newman-Penrose charges that exist in such polyhomogeneous spacetimes [18].
We now consider the non-integrable piece at lower orders. Using the result in (A.13), for the non-integrable piece to vanish at lower orders for general D IJ , it is necessary that it vanishes at the highest order for general D IJ and so s must be an ℓ = 2 spherical harmonic. In this case, applying equation (A.8) to the expression in (3.23), the contribution from D IJ terms at which for general D IJ and s an ℓ = 2 spherical harmonic is not zero. Any further restriction on s will make the integrable piece vanish. There is no need to check / δI (non−int) 3 BC terms at this order since there is no equation linking D IJ to C IJ and B IJ that could result in a cancellation in the non-integrable piece. We deduce that / δI (non−int) 3 D terms is non-vanishing at this order and hence there are no charges at this order, nor subsequent orders as implied by (A.13).
In summary, the complete set of conserved charges obtained at O(r −3 ) are given by (3.21) and (3.35).

Dual BMS charges
We now turn to the tower of dual charges defined in Ref. [21], given by the expression where we have used the form of the background metric of interest (2.2) in the second equality with the 2-form H given by The dual BMS charges can be derived from first principles from the Palatini-Holst action [27,28]. We will consider a 1/r-expansion of the variation of the dual BMS charge The calculations will be analogous to those in Section 3 with similar results being obtained, as with the smooth case [21].
Following Ref. [26], it will be useful to define the twist of a symmetric tensor X IJ Note, if X IJ is trace-free, we can drop the symmetrisation in the definition (4.4). Additionally, it is helpful to note if X and Y are both symmetric trace-free tensors, then Furthermore, if either one of the symmetric tensors X or Y is trace-free, then With the above definitions in mind, equation (4.1) can be written as We now proceed as before to substitute the metric expansions (2.4) and the expression for ξ given in equation (2.40).

Dual charge at O(r 0 )
At leading order, we find  (4.9) which are to be viewed as the generalisation of the NUT charge [26]; see also Ref. [32].

Dual charge at O(r −1 )
At the next order, we find This is analogous to the O(r −1 ) term in Section 3.2, where we found that the charge is zero if strong enough fall-off conditions on the energy-momentum tensor are assumed.

Dual charge at O(r −2 )
At the next order, we find The integrable piece has z degree N D , whereas from (2.38) we deduce that the non-integrable piece has z degree at most N C + 1 ≤ N D . In the case N D > N C + 1, we have charges for n > N C + 1. (4.13) This is analogous to the result in Section 3.3, where we found a set of charges (3.10). As with those charges, the existence of these conserved charges is unsurprising when we consider the Einstein equation (2.34). Thus, as in Section 3.3, the highest non-trivial order is O(z N C +1 ), which we consider next. All terms with z dependence arise from the presence of C IJ , so we start by considering such terms. Assume that T 0m = o(r −4 ), which implies equation (2.27). Rewriting and assuming T mm = o(r −4 ), i.e. equation (2.34), and (2.45) in the first term in (4.14) and (2.43) in the second term, we get where we have used equations (2.34) and (2.45). This is the same expression as was obtained in (3.12), except that the tensor field C IJ has been twisted. Since C IJ is also an arbitrary symmetric, traceless tensor, we again deduce that the highest order term in / δ I 2 (non−int) is zero if and only if D I D J s = 0, i.e. if s is an ℓ = 0 or 1 spherical harmonic. Assuming this to be the case, as we can see from equation (4.15), this implies that the C IJ terms vanish at all orders in z. Furthermore, D I D J s = 0 implies that (2.43) reduces to δB IJ = s∂ u B IJ . Thus, the non-integrable term in equation (4.12) reduces to So as before, there is no non-trivial charge at this order.

Dual charge at O(r −3 )
At the next order, we find that where we have used equations (2.33) and (2.44) to drop all terms involving only C IJ . The integrable piece has z degree N E , whereas (2.38) implies that the non-integrable piece has z degree at most N D ≤ N E . We therefore have a set of conserved charges Once again, this is unsurprising, when we consider the form of Einstein equation (2.35).
Next, we consider the highest order term in / δ I 3 (non−int) and see if it is possible to make this zero in general for a particular choice of s(x I ). The highest order term is O(z N D ), but in the extreme case where N D = N C + 1, it is essential that O(z N C +1 ) terms also vanish. We further assume T 0m = o(r −5 ) and T mm = o(r −5 ). Rewriting where and Note that only the symmetric traceless part of X IJ and Y IJ , and therefore X IJ and Y IJ , need be considered.
The contributions from the X IJ terms and Y IJ terms need to vanish independently in (4.21), as there is no Einstein equation that relates D IJ and C IJ . First, we focus on X IJ .
Use of the Ricci identity and the Schouten identity (B.4) allows us to rewrite X IJ (up to the symmetric, trace-free part) as Then using equation (A.2), we find that the highest order term is So in (4.21), the contribution from X IJ is Henceforth, we assume that s is an ℓ = 2 spherical harmonic satisfying equations (3.31) and (3.32).
Turning our attention to the Y IJ contributions in (4.21), the expression in (4.23) can be rewritten as Since B IJ and C IJ are symmetric and traceless, using Schouten identities (B.5) and (B.6), the second and third lines have zero trace-free symmetric parts and hence can be ignored. The contribution of the Y IJ terms to / δ I 3 (non−int) is therefore simply up to total derivatives. Since B JK is symmetric and traceless, D K D I D J s is projected onto the symmetric trace-free part on its JK indices, which vanishes given that s is an ℓ = 2 spherical harmonic. We conclude that for s an ℓ = 2 spherical harmonic, the O(z N D ) terms in / δ I 3 (non−int) vanish even in the extreme case N C + 1 = N D .
In summary, we have a set of conserved non-trivial charges for m = 0, ±1, ±2.
Reorganising the terms above gives where, we have integrated by parts and used the fact that s is an ℓ = 2 spherical harmonic.
Applying Schouten identities, integrating by parts and applying the equations for an ℓ = 2 spherical harmonic, we obtain The obstruction that prevents an integrable charge existing at this order is exactly the twist of the obstruction in / δI (non−int) 3 D terms in (3.38). In conclusion, the set of conserved charges that can be found by considering / δ I 3 are given by (4.19) and (4.29).

Relating charges to the Newman Penrose Formalism
In this section, we relate the charges obtained here to quantities in the Newman-Penrose formalism [3,18,22]. At O(r −3 ), we will see that the BMS charge and dual charge together form a generalisation of the Newman-Penrose charges for polyhomogeneous spacetimes with finite shear.
The Newman-Penrose formalism begins with a complex null frame {ℓ, n, m,m}, which we choose to be that given in (2.10). Newman-Penrose scalars are then constructed by contracting tensors into null frame components. One such set of complex scalars are the Weyl scalars, given in equation (2.8), which parameterise the ten degrees of freedom of the Weyl tensor. We reproduce these definitions here for convenience Ψ 0 = ℓ a m b ℓ c m d C abcd , Ψ 1 = ℓ a n b ℓ c m d C abcd , Ψ 2 = ℓ a m bmc n d C abcd , Ψ 3 = ℓ a n bmc n d C abcd , Ψ 4 = n amb n cmd C abcd . (5.1) The Riemann tensor is constructed from the Weyl tensor and the Ricci tensor and the ten degrees of freedom of the Ricci tensor, which is constrained by the Einstein equation, are given by three complex and 4 real scalars. The relevant quantities here are both of which are real. Similarly, the connection coefficients may be written in terms of twelve complex scalars. For our purposes, we will only be interested in one such spin coefficient that parameterises the shear of the null congruence generated by the vector field ℓ, All such quantities can be calculated from the metric (2.2), (2.4). We assume that the energy-momentum tensor falls off as T 00 = o(r −5 ), T 0m = o(r −4 ) and T 01 = o(r −3 ). Then one can show that the Weyl scalars fall-off as [19]  which follows from condition (2.16). Furthermore, 5 where the exact values of N 5 and N 6 are unimportant.

Charges at O(r 0 )
At leading order, we obtained the BMS charges and the dual charges in Sections 3.1 and 4.1, Recall that the leading order charges are integrable if and only if ∂ u B IJ = 0. We define a complex quantity In terms of Newman-Penrose quantities, which is conserved if and only if ∂ u σ 0 = 0. This condition is equivalent to ∂ u B IJ = 0; the integrability condition encountered in Sections 3.1 and 4.1.

Charges at O(r −1 )
In sections 3.2 and 4.2, assuming that T mm = o(r −3 ) and T 0m = o(r −4 ), we obtained the following set of integrable charges at the next order 6 Note that the coefficient of each power of z is an independent charge. Letting it can be shown that 14) The first term is trivially conserved since T mm = o(r −3 ) implies ∂ u Ψ 4 0 = 0. Assuming T 01 = o(r −4 ) makes the second term zero. The second term is real and gives the non-trivial conserved charges (3.7) in Section 3.2 when the fall-off of the energy-momentum tensor is not too strong.

Charges at O(r −2 )
At the next order, in Sections 3.3 and 4.3, we obtained the charges 5.16) and showed that the associated non-integrable terms vanished for s an ℓ = 0, 1 spherical harmonic. It can be shown that as an operator equation. Thus, it can be shown that the charges obtained in Sections 3.3 and 4.3 at O(z N C +1 ) and lower can be written in terms of Newman-Penrose quantities as where each coefficient of a z power in Q 2 (z) is an independent conserved charge. Integrating by parts, the differential operators can be moved onto s confirming that this is zero for s an ℓ = 0, 1 spherical harmonic, sinceð 2 Y ℓm = 0 for ℓ = 0, 1.

Charges at O(r −3 )
Finally, and most interestingly, in Sections 3.4 and 4.4, we obtained the charges and showed that the associated non-integrable pieces vanished for s an ℓ = 0, 1 or 2 spherical harmonic. It can be shown that as an operator equation. Thus, Recalling that, furthermore, we have another set of less-interesting conserved charges (3.21) and (4.19), we readily deduce that the expression is a conserved charge for N > N D and any s, including, in particular, when s is an ℓ = 0, 1 or 2 spherical harmonic. The ∂ z terms in (5.24) evaluated at O(z N D ) carry contributions only from charges (5.29) and hence it is possible to produce a more simple expression for the charge built out of Newman-Penrose quantities given by for m = 0, ±1, ±2. (5.30) Integrating by parts, we obtain the generalisation found in Ref. [18] for the Newman-Penrose charges of polyhomogeneous spacetimes with finite shear. For a smooth spacetime, N D = 0; hence the above expression reduces to the original Newman-Penrose charges [22].
In this appendix, we collect some useful properties of polynomials in z. For λ ∈ R {0} and Let p(z) = p n z n + p n−1 z n−1 + O(z n−2 ) be a polynomial in z, then using (A.1), we have by Also, In particular, note that for c independent of z, If we apply a generic linear operator O formed of ∂ z , 1 and ∫ λ with λ = 0 to any p(z) = p n z n + p n−1 z n−1 , we get a new polynomialp(z) = Op(z) of the same degree, which can be expressed in the formp (z) = Ap n z n + Ap n−1 + nBp n z n−1 + O(z n−2 ) (A.5) with A and B n-independent constants depending on the choice of O.
Let X be some tensor of interest, depending on (u, r, x I ) where the r dependence is such that X can be written as a polynomial in z = log r with coefficients depending on (u, x I ), so X = n i=0 X i z i where X i are tensors of the same rank as X and independent of z. Taking angular derivatives of such an expression, for example X, gives an expression of the form where at each order the same function Y appears. Suppose we have an expression P X (z) involving the log r operators above, X and its derivatives, where X only appears linearly. In general, we can decompose such as object as follows where F a (D I )[X] = n i=0 Y a (X i )z i for some Y a . Then by equation (A.5), we can write which implies that equation (A.6) reduces to Note that because we assume X to be some arbitrary tensor, the above equation must hold as an operator equation and should not be viewed an an equation for X n . At the next order Since both X n and X n−1 are arbitrary, these conditions show that if P X (z) vanishes at the highest order, checking that it vanishes at the second highest order only requires one to check that a B a Y a (X n ) = 0 for some arbitrary X n . Furthermore, P X (z) z n−1 = 0 ⇒ P X (z) z n = 0. (A.11) This argument can be extended to all orders, where at each order a new condition arises, but the previous conditions must still be met. We deduce for general X P X (z) z i−1 = 0 ⇒ P X (z) z i = 0 (A.12) for i = 1, ..., n. In particular, considering the contrapositive, P X (z) z i = 0 ⇒ P X (z) z i−1 = 0 ∀ 1 ≤ i ≤ n, (A. 13) i.e. in order for the expression P X (z) to vanish at a particular order for general X, it needs to vanish at all higher orders for general X.
B Identities for tensors on the 2-sphere Schouten identities have been used extensively in this paper to simplify expressions. For a traceless, symmetric tensor X IJ , the Schouten identity implies that [20] ω IJ X KL + ω KL X IJ − ω IL X JK − ω JK X IL = 0.
We use the Ricci identity to exchange the D K and D L derivatives in the last term. This results in an additional term of the form X IK Y J K . Now, taking the symmetric trace-free part of this equation and using equation (B.2) yields identity (B.6).
C ℓ = 0, ℓ = 1 and ℓ = 2 spherical harmonics In this appendix, we list useful properties of ℓ ≤ 2 spherical harmonics. This appendix has a large overlap with appendix C of Ref. [20]. However, given the importance of these results in this paper, for completeness, we reproduce the relevant equations here. If ψ(x I ) is regular, we can assume the expansion (C.1). Plugging this into (C.4) and using the orthogonality relations for spherical harmonics S dΩ Y ℓm Y ℓ ′ m ′ = δ ℓℓ ′ δ mm ′ , we find ℓ m=−ℓ (ℓ − 1)ℓ(ℓ + 1)(ℓ + 2)|ψ ℓm | 2 . (C.5) Notice that each term in the summation on the RHS is non-negative. Therefore, for the RHS to vanish, all terms must individually vanish, which implies that the RHS vanishes if and only if ψ ℓm = 0 for all ℓ > 1, i.e. ψ(x I ) is a linear combination of ℓ = 0 and ℓ = 1 modes. We